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On Page 38, Elementary Set Theory with a Universal Set, Randall Holmes(2012), which can be found here.

We give a semi-formal definition of complex names (this is a variation on Bertrand Russell's Theory of Descriptions):

Definition. A sentence $\psi [(\text{the }y\text{ such that }\phi)/x]$ is defined as $$\begin{align*}&\big((\text{there is exactly one }y\text{ such that }\phi)\text{ implies }(\text{for all }y, \phi\text{ implies }\psi[y/x])\big)\\&\text{ and }\\&\Big(\big(\text{not}(\text{there is exactly one }y\text{ such that }\phi)\big)\text{ implies }\\&\qquad\big(\text{for all }x,(x\text{ is the empty set})\text{ implies }\psi\big)\Big)\;.\end{align*}$$ Renaming of bound variables may be needed.

Definition of the form "$\phi[y/x]$" is:

Definition. When $\phi$ is a sentence and $y$ is a variable, we define $\phi[y/x]$ as the result of substituting $y$ for $x$ throughout $phi$, but only in case there are no bound occurrences of $x$ or $y$ in $\phi$. (We note for later, when we allow the construction of complex names $a$ which might contain bound variables, that $\phi[a/x]$ is only defined if no bound variable of $a$ occurs in $\phi$ (free or bound) and vice versa).

I can't understand why $\psi [($the$\, y\, $such that$\, \phi)/x]$ is defined as it is? Especially, "((not(there is exactly one $y$ such that $\phi$ )) implies (for all $x$, ($x$ is the empty set) implies $\psi$ ))" seems to come out of nowhere.

Feel free to retag this question, I'm not sure if some other disciplines, like elementary set theory, lingusitics are more closely related to it.

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This might help en.wikipedia.org/wiki/On_Denoting –  RParadox Dec 10 '12 at 11:33
    
@RParadox: Thank you for your link. The problem is that it's equally elusive. –  Metta World Peace Dec 10 '12 at 11:46
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2 Answers 2

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Good question. A guess coming up.

General issue: How should we regard expressions of the form "the $\varphi$" or better "the $y$ such that $\varphi(y)$".

Option one: as mere "syntax sugar" that can be parsed away. This is Russell's line. "The $y$ such that $\varphi(y)$" isn't really a complex name, but vanishes on analysis, because (i) $\psi$(the $y$ such that $\varphi(y)$) is equivalent to (ii) there is at least one thing which is $\varphi$ and at most one thing which is $\varphi$ and whatever is $\varphi$ is $\psi$.

Option two: descriptions are complex names. "The $y$ such that $\varphi(y)$" is a complex name of the one and only one thing that is $\varphi$ if there is such a thing, and takes a default value, say the empty set, if there isn't. This was Frege's line.

Both treatments are logically workable. Or we can mix them. Which seems to be what Holmes is doing here.

We do parsing away (a la Russell): but treat the cases where there is and where there isn't a unique $\varphi$ differently, in effect supplying a default value when there isn't (a la Frege). So, roughly speaking, $\psi$(the $y$ such that $\varphi(y)$) says that whatever is $\varphi$ is $\psi$ if there is a unique $\varphi$, but becomes [equivalent to] $\psi(\emptyset)$ when there is no unique $\varphi$.

But I am making this up as I go along, you understand: caveat lector!

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Thank you for your excellent answer. But as a layman, I've no idea how these two competing treatments are "logically workable", which makes me unable to fully understand your argument. Could you please recommend something with a textbook treatment of Russell's and Frege's approaches at an introductory level? –  Metta World Peace Dec 10 '12 at 22:08
    
Metta World Peace, I have tried to do exactly that, i.e. describe the idea of logical analysis. The best thing is to read Frege first, because Russell uses some of his concepts. The most elementary description is his talk on function and concept, see en.wikipedia.org/wiki/Function_and_Concept There are perhaps hundred on textbooks on Frege and Russell, but the best known are two major books of Michael Dummet (philosophy of Language and philosophy of mathematics). –  RParadox Dec 11 '12 at 9:59
    
Usually plato.stanford.edu is a good resource, see plato.stanford.edu/entries/descriptions for example. –  RParadox Dec 11 '12 at 10:02
    
@RParadox: Thank you for your suggestions. –  Metta World Peace Dec 11 '12 at 17:13
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Note: if you want to downvote this, please read it first and then leave a comment.

The way Holmes presents this matter is not clear at all. How is the theory of descriptions and logic related?

A good example is the word "and". We use "and" in the english language and there is the symbol of symbol logic "$ \wedge $". Consider the mapping f from "and" to "$\wedge$" and g from "$\wedge$" to "and".

$f: $ "and" $\rightarrow "\wedge" $

$g: "\wedge" \rightarrow $ "and"

Now, Frege and Russell introduced the symbol logic, so that we can clearly distinguish between "and" and "$\wedge$", because they are not at all the same. Consider the expression "two and two is four". The expression is best translated as 2+2=4. Translation means really taking the first expression and putting into a proper system of logic. For instance here the word "and" was translated into the obivous symbol for addition "+", and not "$\wedge$", although a naive translator would not have known what "and" should stand for.

This matter is not at all trivial, and is not linguistics but logic proper (philosophy if you will). We want to know how the symbols "+" and "$\wedge$" operate. This study is what we call logic in the first place. For instance a few days ago, people downvoted my elementary proof of logic, probably because they thought real mathematicians use lots of strange symbols. However when we are concerned with logic, we can't be so presumptuous. We can't simply throw around symbols and hope that it will all make sense in the end.

Where this kind of analysis comes from is thinking about propositions. What does it mean to talk about anything? Well, if we talk about a thing, we are refering to its existence or non-existence. Which is why a mathematical expression will start with the phrase: $\exists x ...$ or $\nexists x ...$

In his theory of denoting Russells explains why every statements refers to all other things

  1. Definition of all: $C(E) \leftrightarrow \forall x C(x)$
  2. Definition of nothing: $C(N) \leftrightarrow \forall x \neg C(x)$
  3. Definition of exists $C(S)\leftrightarrow \neg \forall x \neg C(x)$

E="Everything",N="Nothing",S="Something". Taken from wikipedia. See also http://en.wikipedia.org/wiki/Theory_of_descriptions

What this achieves is that it shows a certain map, as explained above, but not for "and", but for the expression "exists". So we could say we have explained the map

$h: $ "exists x" $\rightarrow "\exists x"$

, although there are some remaining issues. What one should realize is that all of mathematics essentially is build on this theory, although very mathematicians realize it.

Holmes sentence is an ackward variant of this theory of description. You arrive at it, by applying the given definitions. A much better way to understand the operations is to look at the axioms, see metamath: PL, and play around with them.

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"Holmes sentence is an ackward variant of this theory of description. You arrive at it, by applying the given definitions." A variant, but plainly not arrived at by applying Russell's definitions. Which is why the OP asked the question. –  Peter Smith Dec 10 '12 at 19:32
    
The OP asked the question, because the problem is the definition and the exposition does not explain anything properly. What does it mean to apply predicate logic and what do the notations mean? Shouldn't a book on logic be clear, so that everyone with common sense should be able to follow it? The OP described the analysis of the expression as linguistics (which is actually logic). Perhaps everyone describing logic in this way, should call it something else. The m-logic: reasoning exclusively for mathematicians. Everyone else might want to learn logic. –  RParadox Dec 10 '12 at 20:28
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The OP did not describe the analysis of the expression as linguistics. The OP suggested that the discipline of linguistics might be more closely related to the question than logic is. A book on any subject should ideally be clear. Clarity, however, does not necessarily mean that ‘everyone with common sense should be able to follow it’: common sense is no substitute for an adequate background. –  Brian M. Scott Dec 10 '12 at 22:11
    
This could be called the elitist belief of mathematicians, that their science has to be some kind of black art. However, often, when pressed with a difficult philosophical question, they will not have an answer. Why do we even write $\exists x: \varphi(x)$, what is a function really? and so on. The questions and answers on this site are very representative of these notions. In the end, all these notions depend on certain beliefs. The very idea of logic, is that there is a process of reasoning which can be easily followed. –  RParadox Dec 11 '12 at 9:50
    
If I have a proof, it is expected from me that people who are in the field can understand it and verify it. But if we are talking about logic, there is no such knowledge which can be assumed. Using abstract algebra to prove an elementary theorem in logic is just non-sensical. We are talking about the most fundamental notions, such as those in predicate logic. And anyone who uses complicated language to express elementary concepts is misusing the language. This is certainly true in this case. –  RParadox Dec 11 '12 at 9:54
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